To 3D or not to 3D?
Some of Garmin's recent watches support 3D distance calculations, taking into account the elevation change between recorded points as well as the 2D distance calculated from that latitude and longitude only. The question often comes up of when you should use this? In an ideal world where the location and elevation data were perfectly accurate, the answer would be 'all the time', but in practice, on a level enough route, the elevation part can actually introduce noise and make the overall distance less accurate. By default, nothing on a Fenix 5 has 3D distance on except skiing and snowboarding, where the slopes can be in the region of 45 degrees; that gives a 3D distance (slope length) of √2 times the 2D distance (run length), and at that point, you definitely want to use 3D distance.
So. Where is the switch-over point? The question often comes up for trail runs, which tend to be a bit hillier than your average road marathon might be. I'm going to pick a difference between 2D and 3D of 2% as the cut off to use 3D; I'll discuss why in more detail later.
The key thing to understand is that the relationship between the 3D-2D difference and the slope is very non-linear; for small slopes, the difference is a tiny fraction of a percent, and it's only after you get to about a 20% slope that the difference gets up as far as 2%. Easy maths, it's just Pythagoras' theorem. Here's a graph:
So, for a route that is 20% (either up or down) the whole way, you'd be looking at 2% error if you didn't use 3D distance. Assuming a route which starts and ends at about the same elevation, that would give you an elevation gain which is about 10% of the total distance. That could be quite a brutal course; day one of the Dragon's Back race down Wales is officially 52km with 3800m of ascent, and that's a tough mountain race by most standards. Approximating that as a continuous slope up and down would give you an error of about 1.1%. Still maybe not worth adding 3D distance, especially as that sort of route is probably going to be very hard on GPS anyway and it's unlikely to be as good as 99% accurate in the first place.
But of course, the slope isn't anything like uniform, and because the error isn't linear, there will be areas where the error is much more than 1.1%. Does this matter on average over the route? Perhaps more to the point, are there individual kilometres where it matters, which could mess with your timing?
I looked at two UK routes; the North Downs Way 50 and the Dragon's Back first day mentioned above. I used GPX files of the routes and MapQuest elevation for each point to work out the total 2D and 3D distances for the whole route and individual km, and also calculated the continuous slope approximation described above. (I've run the NDW50, and walked parts of the Dragon's Back route - Crib Goch was actually the first day out for my first pair of walking boots, something of a baptism of fire, as it's a reasonably challenging scramble).
The NDW50 GPX 2D distance came out at 81.325km, which is 50.5 miles. The 3D distance is 81.506, a difference of 0.22%. So, on average, this is still a smaller error than you'd generally find in a GPS track, particularly on a trail route with slopes and tree cover which affect GPS accuracy.
The continuous slope approximation gives 0.098% error, about half the exact value. The error varies from km to km, but only 4 individual km have an error above 1% and only one of those is barely above 2%. So, the very worst km will register as 980m on your watch. Most people will be too busy breathing hard and wondering just how many more flights of wood and mud steps are still to come to notice.
For this route I wouldn't myself use 3D distance; the chances are it'll cause bigger errors than the ones it's there to correct, and even if it corrected the errors perfectly, at 0.22% they're too small compared to typical GPS accuracy to matter. (The biggest error on the day was from missing a route marker, and the distance recorded was 51.27 miles including an extra 0.42 miles from that - a 0.6% difference from the mapped route)
The Dragon's Back might be another story. Here's the elevation profile:
Very, very steep in sections (that last ascent is Crib Goch).
The GPX track distance works out to 47.72km 2D and 48.63km with 3D distance, a total error of 1.9%; the continuous slope approximation is 1.1%, again roughly half the actual error. Even with these elevation changes, the overall error is small enough that, in a mountainous area which is hard on GPS, it might not be your biggest problem. When you look at it km by km, you do see quite large errors where the descent or ascent is particularly steep.
That's to the same scale as the error plot for the NDW50 above, and you can see there's one kilometre with nearly 12% error - your watch would read less than 900m using 2D distance for that km.
Both those routes give errors over the whole route of about twice the continuous slope approximation. So if the CSA gives an error over 1%, the true error might be about 2%, and that's starting to be bigger than cumulative errors in GPS on a reasonably good day. Also, at that point you probably have stretches with slopes long and steep enough that individual kilometres will be badly in error.
So. Where is the switch-over point? The question often comes up for trail runs, which tend to be a bit hillier than your average road marathon might be. I'm going to pick a difference between 2D and 3D of 2% as the cut off to use 3D; I'll discuss why in more detail later.
The key thing to understand is that the relationship between the 3D-2D difference and the slope is very non-linear; for small slopes, the difference is a tiny fraction of a percent, and it's only after you get to about a 20% slope that the difference gets up as far as 2%. Easy maths, it's just Pythagoras' theorem. Here's a graph:
But of course, the slope isn't anything like uniform, and because the error isn't linear, there will be areas where the error is much more than 1.1%. Does this matter on average over the route? Perhaps more to the point, are there individual kilometres where it matters, which could mess with your timing?
I looked at two UK routes; the North Downs Way 50 and the Dragon's Back first day mentioned above. I used GPX files of the routes and MapQuest elevation for each point to work out the total 2D and 3D distances for the whole route and individual km, and also calculated the continuous slope approximation described above. (I've run the NDW50, and walked parts of the Dragon's Back route - Crib Goch was actually the first day out for my first pair of walking boots, something of a baptism of fire, as it's a reasonably challenging scramble).
The NDW50 GPX 2D distance came out at 81.325km, which is 50.5 miles. The 3D distance is 81.506, a difference of 0.22%. So, on average, this is still a smaller error than you'd generally find in a GPS track, particularly on a trail route with slopes and tree cover which affect GPS accuracy.
The continuous slope approximation gives 0.098% error, about half the exact value. The error varies from km to km, but only 4 individual km have an error above 1% and only one of those is barely above 2%. So, the very worst km will register as 980m on your watch. Most people will be too busy breathing hard and wondering just how many more flights of wood and mud steps are still to come to notice.
The Dragon's Back might be another story. Here's the elevation profile:
The GPX track distance works out to 47.72km 2D and 48.63km with 3D distance, a total error of 1.9%; the continuous slope approximation is 1.1%, again roughly half the actual error. Even with these elevation changes, the overall error is small enough that, in a mountainous area which is hard on GPS, it might not be your biggest problem. When you look at it km by km, you do see quite large errors where the descent or ascent is particularly steep.
Both those routes give errors over the whole route of about twice the continuous slope approximation. So if the CSA gives an error over 1%, the true error might be about 2%, and that's starting to be bigger than cumulative errors in GPS on a reasonably good day. Also, at that point you probably have stretches with slopes long and steep enough that individual kilometres will be badly in error.
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